Intrinsic dimensionality and generalization properties of the $\mathcal{R}$-norm inductive bias
This work addresses theoretical limitations of a practical inductive bias in neural networks, which is incremental but clarifies its generalization properties.
The paper investigates the structural and statistical properties of $\mathcal{R}$-norm minimizing interpolants in neural networks, finding that they are intrinsically multivariate and do not achieve statistically optimal generalization in some cases.
We study the structural and statistical properties of $\mathcal{R}$-norm minimizing interpolants of datasets labeled by specific target functions. The $\mathcal{R}$-norm is the basis of an inductive bias for two-layer neural networks, recently introduced to capture the functional effect of controlling the size of network weights, independently of the network width. We find that these interpolants are intrinsically multivariate functions, even when there are ridge functions that fit the data, and also that the $\mathcal{R}$-norm inductive bias is not sufficient for achieving statistically optimal generalization for certain learning problems. Altogether, these results shed new light on an inductive bias that is connected to practical neural network training.